P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In Proc. European Symposium on Algorithms, LNCS vol. 726, pp. 37--48. Springer-Verlag, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of a digraph within a given embedding is described. The algorithm is also able to construct an upward embedding if there exists 3 one. In the variable embedding the upward planarity testing problem is NP complete [10] but it can be solved in polynomial time for digraphs with a single source [4]. In this paper we focus on upward embeddings and orientations of undirected planar graphs. The main contributions of our work are the following: Starting from the properties on upward planarity given in [3] we provide a full characterization of the set of all upward embeddings and ....

P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. SIAM J. Comput., 27(1):132-169, 1998.

....in this intriguing field. What follows is a limited list containing a few examples (a survey on upward planarity can be found in [10] Upward planarity of specific families of digraphs has been studied in: planar st digraphs [14, 7] embedded and triconnected digraphs [3] single source digraphs [13, 4], bipartite digraphs [6] outerplanar digraphs [17] trees [18] and hierarchical digraphs [15] The NP completeness of upward planarity testing is proved in [9] Further, an impressive set of results on upward drawings can be found in the literature on ordered sets. Despite such a long list of ....

P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In Proc. 1st Annu. European Sympos. Algorithms, volume 726 of Lecture Notes Comput. Sci., pages 37--48. SpringerVerlag, 1993.

....drawing such that all th edges are monotone with respect to the vertical direction. It has shown by Garg and Tamassia [9] that recognizing upward planarity of directed graphs is NP complete. However, polynomial algorithms have been developed for testing upward planarity of single source digraphs [4], triconnected digraphs [3] and bipartite graphs [1] Hierarchical graphs are directed graphs with layering structures. A hierarchical graph is conventionally drawn with vertices of a layer on the same horizontal line, and arcs as curves monotonic in vertical direction. A hierarchical graph is ....

P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In Proc. European Symp. on Algorithms, 1993.

....given directed graph (digraph, in short) admits an upward planar drawing, has intrigued researchers for many years. Polynomial time algorithms are known for upward planarity testing of triconnected digraphs [7, 9] bipartite digraphs [35] st digraphs [36, 73] singlesource multiple sink digraphs [64, 10], and outerplanar digraphs [86] However, despite the effort of many researchers, it was not previously known whether one can test a general digraph for upward planarity in polynomial time. Using the concept of a special kind of flow network, we show that the upward planarity testing problem for ....

....digraphs, Thomassen [107] characterizes upward planarity in terms of forbidden circuits. Hutton and Lubiw [64] combine Thomassen s characterization with a decomposition scheme to test upward planarity of a single source digraph in O(n 2 ) time. Bertolazzi, Di Battista, Mannino, and Tamassia [10] show that upward planarity testing of a single source digraph can be done optimally in O(n) time. They also give a parallel algorithm that runs in O(log n) time on a CRCW PRAM with n log log n= log n processors. Di Battista, Tamassia, and Tollis [36, 37] give algorithms for constructing upward ....

P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In 1st Annual European Symposium on Algorithms (ESA '93), volume 726 of Lecture Notes in Computer Science, pages 37--48. Springer-Verlag, 1993.

....Thomassen [Tho89] gave a characterization of upward planarity in terms of forbidden circuits. Later, Hutton and Lubiw [HL91] presented an algorithm to test upward planarity of a single source digraph in quadratic time using Thomassen s result. Bertolazzi, di Battista, Mannino, and Tamassia [BBMT93] showed that upward planarity testing of single source digraphs can be done optimally in linear time. In 1994, Garg and Tamassia [GT95a, GT95b] proved that recognizing upward planarity of directed graphs (digraphs) is NP complete. ffl Recall that (section 1.3) hierarchical graphs are directed ....

P. Bertolazzi, G. di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In Proc. European Symp. on Algorithms, 1993.

.... 68, 82] general graph compute maximal planar subgraph O(n m) Omega Gamma n m) 32, 62, 80, 36] general digraph upward planarity testing NP hard [60] embedded digraph upward planarity testing O(n 2 ) Omega Gamma n) 3] single source digraph upward planarity testing O(n) Omega Gamma n) [4, 69] general graph draw as the intersection graph of a set of unit diameter disks in the plane NP hard [12] It is interesting that apparently similar problems exhibit very different time complexities. For example, while planarity testing can be done in linear time, upward planarity testing is ....

P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In 1st Annual European Symposium on Algorithms (ESA '93), volume 726 of Lecture Notes in Computer Science, pages 37--48. Springer-Verlag, 1993.

....digraphs, Thomassen [28] characterizes upward planarity in terms of forbidden circuits. Hutton and Lubiw [12] combine Thomassen s characterization with a decomposition scheme to test upward planarity of a single source digraph in O(n 2 ) time. Bertolazzi, Di Battista, Mannino, and Tamassia [3] show that upward planarity testing of a single source digraph can be done optimally in O(n) time. They also give a parallel algorithm that runs in O(logn) time on a CRCW PRAM with n log log n= log n processors. Regarding rectilinear planarity testing, Shiloach [24] and Valiant [29] show that ....

P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In Proc. European Symposium on Algorithms, LNCS vol. 726, pp. 37--48. Springer-Verlag, 1993.

....digraphs, Thomassen [22] characterizes upward planarity in terms of forbidden circuits. Hutton and Lubiw [10] combine Thomassen s characterization with a decomposition scheme to test upward planarity of a single source digraph in O(n 2 ) time. Bertolazzi, Di Battista, Mannino, and Tamassia [3] show that upward planarity testing of a single source digraph can be done optimally in O(n) time. They also give a parallel algorithm that runs in O(log n) time on a CRCW PRAM with n log log n= log n processors. Regarding rectilinear planarity testing, Shiloach [18] and Valiant [23] show that ....

P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In 1st Annual European Symposium on Algorithms (ESA '93), Lecture Notes in Computer Science. Springer-Verlag, 1993.

....tree. containing v and f . We also notice that the roots of all but one tree (denoted with T ) are internal vertices of G. Tree T does not contain any internal vertex of G and is rooted at the external face h. The example of Fig. 7 is an illustration of the following theorem. Theorem 4 [5] Let G be an embedded single source digraph, and h a face of G. Digraph G is upward planar, subject to h being the external face, if and only if all the following conditions are satisfied: s (a) c) b) s s Figure 7: a) An embedded single source digraph G. b) The face sink graph ....

....Single Source Digraphs Hutton and Lubiw [20] show how to apply Theorem 3 to test the upward planarity of an embedded single source digraph with n vertices in O(n 2 ) time. Theorem 4 yields the following algorithm for testing the upward planarity of an embedded single source digraph G [5]. 1. Test whether G is acyclic and bimodal. If either of these properties is not verified, return not upward planar . 2. Construct the face sink graph F of G (see Section 3.4) 3. Check Conditions 2 and 3 of Theorem 4. If these conditions are not verified, then return not upward planar , ....

[Article contains additional citation context not shown here]

P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In 1st Annual European Symposium on Algorithms (ESA '93), volume 726 of Lecture Notes in Computer Science, pages 37--48. SpringerVerlag, 1993.

....digraphs, Thomassen [33] characterizes upward planarity in terms of forbidden circuits. Hutton and Lubiw [14] combine Thomassen s characterization with a decomposition scheme to test upward planarity of a single source digraph in O(n 2 ) time. Bertolazzi, Di Battista, Mannino, and Tamassia [5] show that upward planarity testing of a single source digraph can be done optimally in O(n) time. They also give a parallel algorithm that runs in O(log n) time on a CRCW PRAM with n log log n= log n processors. Di Battista, Tamassia, and Tollis [10, 11] give algorithms for constructing upward ....

P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In 1st Annual European Symposium on Algorithms (ESA '93), volume 726 of Lecture Notes in Computer Science, pages 37--48. Springer-Verlag, 1993.

....Thomassen [52] characterizes upward planarity in terms of forbidden circuits. Hutton and Lubiw [31] combine Thomassen s characterization with a decomposition scheme to test upward planarity of an n vertex single source digraph in O(n 2 ) time. Bertolazzi, Di Battista, Mannino, and Tamassia [5] give optimal algorithms for upward planarity testing of single source digraphs: Upward planarity testing of a single source digraph with n vertices can be done in O(n) time, and in O(logn) time on a CRCW PRAM with n log log n= log n processors. Garg and Tamassia [29] have recently settled ....

P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In 1st Annual European Symposium on Algorithms (ESA '93), Lecture Notes in Computer Science. Springer-Verlag, 1993.

No context found.

P. Bertolazzi, G. Di Battista, C. Mannino, and R. Tamassia. Optimal upward planarity testing of single-source digraphs. In Proc. European Symposium on Algorithms, LNCS vol. 726, pp. 37--48. Springer-Verlag, 1993.

No context found.

Paola Bertolazzi, Giuseppe Di Battista, Carlo Mannino, and Roberto Tamassia. Optimal upward planarity testing of single-source digraphs. SIAM Journal on Computing, 27(1):132--196, 1998.

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